Stanley–Stembridge conjecture for the functions $E^{\nu'}_{\lambda/\mu}$

Let $\lambda$ and $\mu$ be partitions with $\mu\subseteq\lambda$, and let $E^{\nu'}_{\lambda/\mu}$ be the symmetric functions obtained by expanding the associated Frobenius character in the Schur basis in the $\mathbf{x}$ variables. A symmetric function is $h$-positive if it is a nonnegative linear combination of complete homogeneous symmetric functions. Stanley–Stembridge's conjecture. The symmetric functions

$$E^{\nu'}_{\lambda/\mu}$$

are $h$-positive for every $\lambda$, $\mu$ and $\nu'$. This is stated as an equivalent formulation of the monomial-immanant conjecture; the source gives no resolution.